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Gödel's incompleteness theorems first appeared together in a paper titled (translated to English) "On formally undecidable propositions of Principia Mathematica and related systems I," with the Roman numeral at the end of the title indicating that this was to be the first paper of a series to be published. The only explicit mention of subsequent papers I've found is in Torkel Franzén's book on the incompleteness theorems which mentions (only in passing) that a second paper was planned but never written.

My question is: what was the planned subject of this second paper? I'm intrigued by the thought that such fundamental work was only the first half of a planned corpus of research. In addition, if someone happens to know why the paper was never written, I'd be interested in learning about why.

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You can see the article dedicated to Kurt Gödel into Stanford Encyclopedia of Philosophy, by Juliette Kennedy :

With regard to the Second Incompleteness Theorem, the argument relies in part on formalizing the proof of the First Incompleteness Theorem as we saw. This step is omitted in Gödel 1931. He planned to include the step in what would have been a second part II [... (emphasys added)]. But instead of writing it he turned to the continuum problem.

(Part II was to elaborate on other points too: the ‘true reason for incompleteness,’ and the applicability of the two theorems to other systems.) He perhaps did not feel compelled to attend to what looked like an exercise in formalization, relying instead on the informal argument to convince (in which it succeeded). However this step turned out to be somewhat non-trivial. [...]

Eventually a complete proof of the Second Theorem was given by Hilbert and Bernays in some seventy pages in their Hilbert and Bernays 1939. A much more compact treatment of the theorem was given by Löb in his Löb 1956, and subsequently Feferman, in his 1960 “Arithmetization of Metamathematics in a General Setting”.

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    $\begingroup$ I read somewhere (I forget where) that Godel had only intended the second paper because he suspected no one would believe the result in the first one without it; when the finding of the first paper was widely accepted, he didn't feel the need. But as I said, I can't remember the source, so perhaps it's apocryphal. It's an entertaining piece of apocrypha in any case. $\endgroup$
    – Malice Vidrine
    Feb 22, 2014 at 21:39

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